Explain on what factors the ionization enthalpies of elements having $d^{n}$ configuration depend upon? Explain the variation in ionization enthalpies of transition elements in the $3d$ series.

(N/A) The ionization enthalpy depends on three factors:
$(i)$ Nucleus-electron attraction
$(ii)$ Electron-electron repulsion
$(iii)$ Exchange energy
Exchange energy is responsible for the stabilization of the energy state. It is approximately proportional to the total number of possible pairs of parallel spin degenerate orbitals. When several electrons occupy a set of degenerate orbitals, the lowest energy state corresponds to the maximum possible extent of single occupation of orbitals and parallel spins (Hund's Rule). The loss of exchange energy increases the stability and hence the ionization enthalpy increases.
$(i)$ First ionization enthalpy: The first ionization enthalpy follows an irregular trend. This is because the first electron when removed alters the relative energies of $3d$ and $4s$ orbitals. The electrons are first removed from $4s$ and then from $3d$ orbitals.
In the first transition series, going from scandium to zinc, the nuclear charge increases with the increase in atomic number and the electrons are added in $3d$ orbitals. The increase in nuclear charge is opposed by the shielding effect of $3d$ electrons; as a result, the atomic radii decrease less rapidly and so there is a slight increase in ionization enthalpy in the $3d$ series.
$(ii)$ Successive ionization enthalpies: In general, the third ionization enthalpy is higher than the second, which is higher than the first. The high values of successive ionization enthalpies are attributed to high effective nuclear charge and poor shielding of one $d$-electron by another.
The second ionization enthalpy of chromium is higher than manganese, while the third ionization enthalpy of manganese is higher than chromium. In the case of chromium, the second electron is to be removed from a half-filled $(d^{5})$ subshell which is extra stable. In the case of manganese, the third electron is to be removed from a half-filled $(d^{5})$ subshell, making the third ionization enthalpy higher for manganese than chromium.
${}_{24}Cr^{+}:[Ar] 3d^{5} 4s^{0} \rightarrow {}_{24}Cr^{2+}:[Ar] 3d^{4}$
${}_{25}Mn^{2+}:[Ar] 3d^{5} 4s^{0} \rightarrow {}_{25}Mn^{3+}:[Ar] 3d^{4}$ (More difficult to attain)
Thus, for manganese, it is difficult to remove the third electron.
$(iii)$ Stability of $M^{2+}$ ions in gaseous state: The stability of $M^{2+}$ ions depends on the sum of the first and second ionization enthalpies and the enthalpy of atomization. Lower the sum, greater the thermodynamic stability. The dominant factor is the second ionization enthalpy. This explains why $Zn^{2+}$ and $Mn^{2+}$ ions are formed easily and the removal of a third electron is difficult.